Question

find ( f(-2) ) if ( f(x)=\frac{x^{3}}{7}-8x ).

( f(-2)=square ) (simplify your answer. type an integer or a fraction.)

Explanation:

Step1: Differentiate the function

Use the power rule \((x^n)^\prime = nx^{n - 1}\).
For \(y=\frac{x^{3}}{7}-8x\), the derivative \(y^\prime=\frac{3x^{2}}{7}-8\).

Step2: Substitute \(x = - 2\)

Substitute \(x=-2\) into \(y^\prime\).
\(y^\prime|_{x = - 2}=\frac{3\times(-2)^{2}}{7}-8\).
First calculate \((-2)^{2}=4\), then \(\frac{3\times4}{7}-8=\frac{12}{7}-8\).
Rewrite \(8\) as \(\frac{56}{7}\), so \(\frac{12}{7}-\frac{56}{7}=\frac{12 - 56}{7}\).
\(12-56=-44\), then \(\frac{-44}{7}\).

Answer:

\(-\frac{44}{7}\)